Dynamic Patterns of GeneRegulation I: Simple Two-GeneSystemsStefanie WidderJosef SchichoPeter Schuster

SFI WORKING PAPER: 2006-03-007

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SANTA FE INSTITUTE

Dynamic Patterns of Gene Regulation I:

Simple Two Gene Systems

Stefanie Widdera, Josef Schichob, and Peter Schuster∗ a,c

a Institut fur Theoretische Chemie der Universitat Wien,

Wahringerstraße 17, A-1090 Wien, Austria,

b RICAM – Johann Radon Institute for Computational and Applied

Mathematics of the Austrian Academy of Sciences, Altenbergerstraße 69,

A-4040 Linz, Austria, and

b Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA

Abstract: Regulation of gene activities is studied by means of a newcomputer assisted mathematical analysis of ordinary differential equations(ODEs) derived from binding equilibria and chemical reaction kinetics.Here we present results on cross-regulation of two genes through activa-tor and/or repressor binding. Arbitrary (differentiable) binding functioncan be used but systematic investigations are presented for gene-regulatorcomplexes with integer valued Hill coefficients up to n = 4. The dynamicsof gene regulation is derived from bifurcation patterns of the underlyingsystems of kinetic ODEs. In particular, we present analytical expressionsfor the parameter values at which one-dimensional (transcritical, saddle-node or pitchfork) and/or two-dimensional (Hopf) bifurcations occur. Aclassification of regulatory states is introduced, which makes use of he signof a ’regulatory determinant’ D (being the determinant of the block inthe Jacobian matrix that contains the derivatives of the regulator bindingfunctions): (i) Systems with D < 0, observed, for example, if both pro-teins are activators or repressors, give rise to one-dimensional bifurcationsonly and lead to bistability for n ≥ 2, and (ii) systems with D > 0, foundfor combinations of activation and repression, sustain a Hopf bifurcationand undamped oscillations for n > 2. The influence of basal transcriptionactivity on the bifurcation patterns is described. Binding of multiple sub-units can lead to richer dynamics than pure activation or repression statesif intermediates between the unbound state and the fully saturated DNAinitiate transcription. Then, the regulatory determinant D can adopt bothsigns, plus and minus.

Key words: Basal transcription – bifurcation analysis – cooperative bind-ing – gene regulation – Hill coefficient – Hopf bifurcation.

∗Corresponding author, E-mail: pks@tbi.univie.ac.at

1

Notation

metabolites A,B,C, . . . ,

concentrationsa of metabolites [A] = a, [B] = b, [C] = c . . . ,

genes G1,G2,

concentrationsa of genes [G1] = g1, [G2] = g2,

transcribed (m)RNAs Q1,Q2,

concentrations of RNAs [Q1] = q1, [Q2] = q2,

translated proteins P1,P2,

concentrationsa of proteins [P1] = p1, [P2] = p2,

gene-protein complexes G1 · P2 = H1, G2 · P1 = H2,

concentrationsa of complexes [G1 · P2] = [H1] = h1,

[G2 · P1] = [H2] = h2,

dissociation constants K1 = [G2]·[P1][H2]

= g2·p1

h2

,

K2 = [G1]·[P2][H1]

= g1·p2

h1

,

transcription rate constants kQ

1 , kQ

2 ,

translation rate constants kP

1 , kP

2 ,

RNA degradation rate constants dQ

1 , dQ

2 ,

protein degradation rate constants d P

1 , d P

2 ,

binding functions F1(p2) , F2(p1) ,

ratios of rate constants ϑ1 =kQ1·kP

1

d Q1·d P

1

, ϑ2 =kQ2·kP

2

d Q2·d P

2

,

φ1 =d P1

kP1

, φ2 =d P2

kP2

,

regulatory determinant D(p1, p2) =

= − kQ

1 kQ

2 kP1kP

2

∣∣∣∣∣∣0

(∂F1

∂p2

)

(∂F2

∂p1

)0

∣∣∣∣∣∣.

aDepending on conditions the symbols express concentrations or activities.

2

3

1 Introduction

Theoretical work on gene regulation goes back to the nineteen sixties [1]soon after the first repressor protein had been discovered [2]. A little laterthe first paper on oscillatory states in gene regulation was published [3]. Theinterest in gene regulation and its mathematical analysis never ceased [4–6]and saw a great variety of different attempts to design models of geneticregulatory networks that can be used in systems biology for computer simu-lation of gen(etic and met)abolic networks.1 Most models in the literatureaims at a minimalist dynamic description which, nevertheless, tries to ac-count for the basic regulatory functions of large networks in the cell in orderto provide a better understanding of cellular dynamics. A classic in gen-eral regulatory dynamics is the monograph by Thomas and D’Ari [7]. Thecurrently used mathematical methods comprise application of Boolean logic[8–10] stochastic processes [11] and deterministic dynamic models (examplesare [12–14]. In vivo constructs and selection experiments [15–19] provideinsight into regulatory dynamics and better understanding of genabolic net-works. Apart from diverse minimalist models [20], relatively few articles areconcerned with the mechanistic prerequisites for the occurrence of certaindynamic features based on positive and negative feedback loops [21, 22] likestability, bistability, periodicity or homeostasis. Only few essentially newresults are presented in this contribution. Instead we carry out the analyt-ical approach further than in other papers and present a new approach tobifurcation analysis that allows for classification of the dynamical systemsfor gene regulation.

The basic gene regulation scenario that underlies the calculations pre-sented here is sketched in figure 1 and has been adopted from the bookletby Ptashne & Gann [23]. Two classes of molecular effectors, activators andrepressors, decide on the transcriptional activity of a gene, whose activity isclassified according to three states: (i) ‘Naked’ DNA is commonly assumedto have a low or basal transcription activity (basal state), (ii) transcriptionrises to the normal level when (only) the activator is bound to the regula-tory region of the gene (active state), and (iii) complexes with repressor areinactive no matter whether the activator is present or not (inactive state).The basal state is sometimes also characterized as ’leaky transcription’. Weshall use this notion here for a general term in the kinetic equations that de-scribes unregulated transcription. Effectors often become active as oligomers,commonly dimers or tetramers, and therefore we shall refer also cases wheremore than one molecule have to bind before regulation becomes effective (Foran overview of mathematical approaches to various binding equilibrium thatare of relevance in gene regulation see [24]). The genetic regulatory systemis completed by translation of the transcribed mRNAs into protein regula-tors. Both classes of molecules, mRNAs and proteins, undergo degradationthrough a first order reaction. DNA, in the form of the genes is assumed to bepresent at constant concentration. Transcription, translation, and degrada-

1Discussion and analysis of combined genetic and metabolic networks has be-come so frequent and intense that we suggest to use a separate term, genabolic

networks for this class of complex dynamical systems.

4

Figure 1: Basic principle of gene regulation. The figure sketches the reg-

ulated recruitment mechanism of gene activity control in prokaryote cells as dis-covered with the lac genes in Escherichia coli [23]. The gene has three states ofactivity, which are regulated by the presence or absence of glucose and lactose inthe medium: State I, basal state called ’leaky transcription’ occurs when bothnutrients are present and it is characterized by low level transcription; neither theactivator, the cap protein, nor the lac-repressor protein are bound to their siteson DNA. State II, activated state is induced by the absence of glucose and thepresence of lactose and then cap is bound to DNA, but lac-repressor protein isabsent. Finally, when lactose is absent the gene is in the inactive state no mat-ter whether glucose is available or not. Then, the lac-repressor protein is boundto DNA and transcription is blocked. The promotor region of the DNA carriesspecific recognition sites for the RNA polymerase in addition to the binding sitesfor the regulatory proteins.

5

tion are multi-step processes and follow rather involved reaction mechanisms.A carefully studied example of such a multi-step process is template-inducedRNA synthesis commonly called plus-minus RNA replication [25, 26]. In casemonomers and enzyme, the bacteriophage Qβ replicase, are present in excess,the over-all kinetics, however, follows simple first-order rate laws. We shalladopt simple kinetic first order expressions for transcription and translationhere.

Following our approach gene regulatory systems can be grouped into twoclasses: (i) Simple systems for which a complete (computer assisted) quali-tative analysis can be carried out analytically,2 and (ii) complex systems forwhich qualitative analysis is pending because of hard computational problemsor principal difficulties. In both classes the binding function may be arbitrar-ily complicated provided it is differentiable. The distinction between the twoclasses is made in section 3.2 by means of the Jacobian matrix of the dynam-ical systems. In particular, all cross-regulatory two gene systems are of class(i) no matter how sophisticated the binding functions are. In a forthcomingstudy [27] we shall present analogous results for cases in which the analysisin more involved. These systems include two gene systems where the geneshave double functions, for example self-repression and cross-activation, andregulatory systems with more than two genes apart from those with cyclicregulation (1 → 2, 2 → 3, . . . , n → 1) which fall also into class (i). Here wepresent the analysis of the ODEs derived from chemical reaction kinetics ofgene regulation under the assumption of fast binding equilibria. The (com-puter assisted) qualitative analysis of the dynamical systems is followed bya discussion of results obtained for some special cases with Hill coefficientsn = 1, 2, 3, and 4.

2 Kinetic equations

2.1 Binding equilibria

The DNA is assumed to carry two genes, G1 and G2, which have bindingsites for effectors, activators and/or repressors in the promoter region. Bind-ing of the proteins is assumed to occur fast compared to transcription andtranslation, and accordingly the equilibrium assumption is valid. The binaryinteraction is restricted to cross-regulation of the two genes: The translationproduct of gene G1 controls the activity of gene G2 and vice versa. In otherwords, the activity of gene G1 is a function of the equilibrium concentrationof protein P2, p2, and gene G2 is likewise controlled by P1:

G1 + n2 P2

g0 F1(p2;n2,...)−−−−−−−−−⇀↽−−−−−−−−− G1 · P2 and (1)

G2 + n1 P1

g0 F2(p1;n1,...)−−−−−−−−−⇀↽−−−−−−−−− G2 · P2 . (2)

Since the number of DNA molecules is constant, both genes are present inthe same total concentrations: (g1)0 = (g2)0 = g0. In the simplest case,

2Computer assistance in simple problems may involve computation of solutionsfor equations but does not require full simulations of regulatory dynamics.

6

binding equilibria of monomers, n1 = n2 = 1, and mass action we obtain:3

G1 + P2

K−1

2

G1 · P2 and (3)

G2 + P1

K−1

1

G2 · P1 . (4)

The concentration of the gene protein complex is expressed by

[G1 · P2] = c1 = g0 ·p2

K2 + p2

≈ g0 ·(p2)0

K2 + (p2)0

[G2 · P1] = c2 = g0 ·p1

K1 + p1

≈ g0 ·(p1)0

K1 + (p1)0

,

where we approximate the equilibrium protein concentrations by the totalconcentrations, p1 ≈ (p1)0 and p2 ≈ (p2)0, assuming that the numbers ofgenes are smaller than the numbers of effector molecules. In order to formu-late cross-regulation of two genes in versatile form we generalize the dimen-sionless regulatory functions, Fj, j = 1, 2, to cooperative interactions witharbitrary exponents n (see section 4.2):

Gene “1”

{F

(act)1 (p2; K2, n) =

pn

2

K2 + pn

2

activation,

F(rep)1 (p2; K2, n) = K2

K2 + pn

2

repression,

Gene “2”

{F

(act)2 (p1; K1, n) =

pn

1

K1 + pn

1

activation,

F(rep)2 (p1; K1, n) = K1

K1 + pn

1

repression .

(5)

Here ‘rep’ and ‘act’ stand for repression and activation, respectively, whereeither the free gene, Gj, or the complex, Gj ·Pi, are transcribed. The exponentn, in particular when determined experimentally, is called the Hill coefficient(See [28] and [29], p. 864 ff.). The Hill coefficient n is related to the molecularbinding mechanism. In simple cases n is the number of protein monomersrequired in binding to the DNA in order to achieve effector activity.

More than one parameter will be required for binding equilibria thatinvolve more than one protein subunit. To give an example, consecutivebinding of four ligands P2 to gene G1,

G1 + 4P2

K−1

21

H(1)1 + 3P2

K−1

22

H(2)1 + 2P2

K−1

23

H(3)1 + P2

K−1

24

H(4)1 ,

where H(k)1 = G1 · (P2)k, the complex formed by the gene with k protein

monomers. If the only complex that is active in transcription were H(4)1 the

binding function would adopt the form4

F(act)1 (p2; K21, . . . , K24) =

=p4

2

K21K22K23K24 + K22K23K24 p2 + K23K24 p22 + K24 p3

2 + p42

.

3It will turn out that the usage of dissociation rather than binding constants isof advantage and therefore we define K = [G] · [P]/[G · P].

4The equilibrium constants applied are macroscopic dissociation constants. Forequivalent microscopic constants the individual terms in the denominator receivethe binomial coefficients, (1, 4, 6, 4, 1), as factors.

7

Examples of different binding functions will be discussed together with theresults derived for the individual systems.

2.2 Reaction kinetics

The transcription reactions come in two variants, an activating mode (cor-responding to state II of figure 1) and a repressing mode (corresponding tostate III of figure 1). The basal state (state I) can be included in the ac-tivating or the repressing mode as we shall see later. The kinetic reactionmechanism for transcription then has the following form:

G1 · P2

kQ1−−−→ G1 + Q1 activation,

G1

kQ1−−−→ G1 + Q1 repression,

(6)

G2 · P2

kQ2−−−→ G2 + Q2 activation,

G2

kQ2−−−→ G2 + Q2 repression.

(7)

In case of activation, the regulator-gene complexes are transcribed, whereasthe complexes are inactive in repression and transcription is mediated by thefree genes.

In contrast to DNA, the transcription products, the mRNAs Q1 and Q2,as well as the regulators, the proteins P1 and P2, have only finite lifetimebecause of decay reactions. For translation of mRNAs and for degradationof mRNAs as well as proteins we find:

Qi

kPi−−−→ Qi + Pi , i = 1, 2 : translation , (8)

Qi

d Qi−−−→ 0 , i = 1, 2 : degradation , and (9)

Pi

d Pi−−−→ 0 , i = 1, 2 : degradation . (10)

Translation and degradation reactions are modelled as simple single step pro-cesses. The approximation for translation is well justified in case of excessmonomers, ribosomes and other translation factors (as mentioned in the in-troduction). Simple degradation reactions are always of first order. Sincethe total concentration of the genes is constant and since we shall applyonly binding functions that are proportional to g0, we can absorb the DNA

concentration in the rate constant for transcription: kQ

i = kQ

i · g0. As a con-sequence the rate parameters have different dimensions, [kQ

i ] = [m× t−1] and[kP

i ] = [dQ

i ] = [d P

i ] = [t−1] where m stands for ’molar’ and t stands for ’time’.These substitutions are advantageous in a second aspect, too: The regu-latory functions are dimensionless, no matter whether we are using simplehyperbolic or higher order binding equilibria.

Now, we are in a position to write down the kinetic differential equations

8

for all four molecular species, Q1, Q2, P1, and P2, derived from two genes:

dqi

dt= qi = kQ

i Fi(pj) − dQ

i qi , i = 1, 2 , j = 2, 1 , and (11)

dpi

dt= pi = kP

i qi − d P

i pi , i = 1, 2 . (12)

Accordingly, the dynamical system contains eight kinetic parameters andtwo binding functions. Except for the binding functions Fi(pj) the systemis linear. This property will be important for analyzing the Jacobian matrixand determining the stability of stationary points.

3 Qualitative analysis

3.1 Determination of stationary points

In order to derive equations for the stationary or fixed points of the dynamicalsystem (11,12) we introduce four ratios of reaction rate parameters,

ϑi =kQ

i kP

i

dQ

i d P

i

and φi =d P

i

kP

i

, i = 1, 2 , (13)

that simplify the expressions obtained from qi = pi = 0 , i = 1, 2:

pi − ϑi Fi(pj) = 0 , i = 1, 2 , j = 2, 1 . (14)

The binding functions are normalized 0 ≤ Fi ≤ 1 and hence the equilibriumconcentrations of proteins are confined to values in the range 0 ≤ pi ≤ϑi with i = 1, 2. Commonly, the binding functions Fi are ratios of twopolynomials and then, equation (14) can always be transformed into twocoupled polynomials. Examples will be given in the forthcoming sections.Provided the stationary values of the protein concentrations are known wefind for the mRNAs

qi = φi pi , i = 1, 2 . (15)

Again we point at a difference in dimensions: [ϑi] = [m], whereas the φi’s aredimensionless. Stationary concentrations are completely defined by the tworatios of kinetic constants, ϑ1 and ϑ2 (and, of course, by the parameters inthe functions F1 and F2). According to equation (15) the stationary mRNAconcentrations qi are related to the corresponding protein concentrations pi

through multiplication by a positive factor and hence qi is zero if and only ifpi = 0.

Apart from an initial phase determined largely by the choice of theinitial values qi(0), pi(0) (i = 1, 2) the projection of the trajectories(q1(t), q2(t), p1(t), p2(t)

)onto the (q1, q2)-subspace shows close similarity to

that onto the (p1, p2)-plane. For stability analysis it is sufficient therefore toconsider the fixed points and their properties on either of the two subspaces.We choose the ’protein’-subspace, P = {pi; pi ≥ 0∀ i = 1, 2} since proteinconcentrations are calculated more directly. It is worth noticing that thepositions of the stationary points, P = (p1, p2) ∈ P, depend only on ϑ1 and

9

Table 1: Protein concentrations in the strong and weak binding limits.

The limits were calculated from equations (17-19) by taking the limits limK1 → 0and/or limK2 → 0 or limK1 → ∞ and/or limK2 → ∞, respectively.

Strong binding: Weak binding:

System lim Kj → 0 lim Kj → ∞j p1 p2 j p1 p2

act-acta 1 0 0 1 0 0

ϑ1ϑn

K2+ϑn

2

ϑ2

2 0 0 2 0 0

ϑ1 ϑ2ϑn

K1+ϑn

1

1,2 0 0 1,2 0 0

ϑ1 ϑ2

act-rep 1 0 0 1 ϑ1ϑn

2

K2+ϑn

2

ϑ2

2 ϑ1 ϑ2K1

K1+ϑn

1

2 0 ϑ2

1,2 0 0 1,2 0 ϑ2

rep-rep 1 ϑ1 0 1 ϑ1K2

K2+ϑn

2

ϑ2

2 0 ϑ2 2 ϑ1 ϑ2K1

K1+ϑn

1

1,2 ϑ1 0 1,2 ϑ1 ϑ2

0 0

0 ϑ2

a The solution (p1 = 0, p2 = 0) is a double root in the strong binding limit.

ϑ2 and not on all eight kinetic parameters. Substitution of p2 = ϑ2F2(p1)yields the solution:

p1 − ϑ1 F1

(ϑ2 F2(p1)

)= 0 . (16)

Equation (16) leads to high order polynomials for nonlinear binding functionswhich, nevertheless, are computed straightforwardly for general n for thesimple binding functions (5). For activation-activation, activation-repression,

10

and repression-repression, we obtain

p1 ·(

pn·n1 ϑn

2 − pn·n−11 ϑ1ϑ

n2 + K2 ·

n∑

k=0

pn·(n−k)1

(n

k

)Kk

1

)= 0 , (17)

K2 ·(

n∑

k=0

pn·(n−k)+11

(n

k

)Kk

1

)+ (p1 − ϑ1) · (ϑ2K1)

n = 0 , and (18)

(p1 − ϑ1) · K2 ·(

n∑

k=0

pn·(n−k)1

(n

k

)Kk

1

)+ p1 · (ϑ2K1)

n = 0 , (19)

respectively. The equilibrium concentration p2 is readily obtained from

p2 =ϑ2 · pn

1

K1 + pn1

for (17) and p2 =ϑ2 · K1

K1 + pn1

for (18) and (19).

From equation (17) follows that the origin is always a fixed point foractivation-activation systems, P1 = (0, 0), corresponding to both genes si-lenced. The degree of the polynomials in p1, πn = n2 + 1, increases with thesquare of the Hill coefficient and thus reaches already 17 for n = 4. Neverthe-less, we obtained never more than three or four real roots through numericalsolution (for n ≤ 4). Obviously, we have an even number of real roots for nodd (1 and 3), and an odd number of real roots for n even (2 and 4).

The high degree of the polynomials is prohibitive for direct calculationsbased on the equations (17-19) but the expressions are suitable to compute,for example, the limits of the equilibrium concentrations for strong and weakbinding, lim K1,2 → 0 and lim K1,2 → ∞, respectively. The results are showin table 1 and they correspond completely to the expectations. In case thelimits are taken for both constants simultaneously the limiting concentra-tions are independent of the Hill coefficient n – not unexpectedly since allfunctions Fi(pj; Kj, n) approach either zero or one in these limits. Examplesof individual dynamical systems will be discussed in section 4 and thereforewe mention only one general feature here: In the strong binding limit thecombination activation-activation leads to two active genes or two silencingof both genes, whereas we have alternate activities – ‘1’ active and ‘2’ silentor ‘1’ silent and ‘2’ active – in the repression-repression system. Weak bind-ing, on the other hand, silences the genes in the act-act case and leads tofull activities in rep-rep systems.

In the next subsection 3.2 we shall make use again of equations (17-19)and derive limits of functions for the strong binding case, which are appliedto the analysis of the regulatory dynamics in parameter space.

3.2 Jacobian matrix

The dynamical properties of the ODEs (11,12) are analyzed by means of theJacobian matrix and its eigenvalues. For the combined vector of all variables,x = (x1, . . . , x4) = (q1, q2, p1, p2), the Jacobian matrix A has a useful block

11

Figure 2: Eigenvalues of the Jacobian matrix (20). The four eigenvalues ofa two-gene system, ε1, ε2, ε3, and ε4, are plotted as functions of D around thepoint D = 0 as reference. The dimension of the ordinate axis is reciprocal time,[t−1]. At D = 0 and different values of dQ

1 , dQ

2 , d P

1 and d P

2 we observe four negativereal eigenvalues of the Jacobian, which are turning into complex conjugate pairsat the values D = D1, D = D2, and D = D3. At DoneD and at DHopf the fixedpoint changes stability. The one dimensional bifurcation lies at negative values ofD since DoneD < 0, whereas DHopf > 0, and thus the Hopf bifurcation appearsalways at positive D-values. Color code: Real eigenvalues are drawn in black andthe real parts of complex conjugate pairs of eigenvalues are shown as red lines.

structure:

A =

{aij =

∂xi

∂xj

}=

Qd

... Qk. . . . . . . . . .Pk

... Pd

=

=

−dQ

1 0... kQ

1∂F1

∂p1

kQ

1∂F1

∂p2...0 −dQ

2

... kQ

2∂F2

∂p1

kQ

2∂F2

∂p2.... . . . . . . . . . . . . . . . . . . . . . . . . . . ....kP

1 0... −d P

1 0...0 kP

2

... 0 −d P

2

. (20)

This block-structure of matrix A largely facilitates the computation of the 2neigenvalues [30, 31]. Since the matrices Qd and Pk commute, Qd·Pk = Pk·Qd,the relation

∣∣∣∣Qd Qk

Pk Pd

∣∣∣∣ = |Qd · Pd − Qk · Pk |

holds. In certain cases, among them all forms of cross-regulation of twogenes and all forms of cyclic pairwise regulation of more than two genes,Gn ⇒ G1 ⇒ G2 ⇒ · · · ⇒ Gn for arbitrary n [27], the secular equation is of

12

the form:5

(ε + dQ

1 )(ε + dQ

2 )(ε + d P

1 )(ε + d P

2 ) + D = 0 with

D = − kQ

1 kQ

2 kP

1kP

2 Γ(p1, p2) .(21)

Since D and Γ(p1, p2) are obtained from the binding functions by calculatingthe derivatives with respect to the protein concentrations,

Γ(p1, p2) = −∣∣∣∣∣

0 ∂F1

∂p2

∂F2

∂p1

0

∣∣∣∣∣ =∂F1

∂p2

· ∂F2

∂p1

, (22)

we call D the regulatory determinant of the dynamical system. Knowl-edge of D is sufficient to analyze the stability of fixed points and to cal-culate the parameter values at the bifurcation points. At D = 0 the eigen-values of the Jacobian are the set of all 4 negative degradation rate con-stants, −dQ

i and −d P

i (i = 1, 2), which we assume to be ordered by value:ε1 = −min{dQ

1 , dQ

2 , d P

1 , d P

2 } is the largest and ε4 = −max{dQ

1 , dQ

2 , d P

1 , d P

2 }is the smallest eigenvalue of A. In the non-degenerate case, i.e. when alldegradation rate parameters are different, the eigenvalues correspond to fourpoints on the negative (reciprocal time) axis represented by the ordinate axisin figure 2. For a fixed point P ∈ P with D = 0 this implies asymptoticstability. Non-generic cases with double or multiple real roots at D = 0 im-ply also asymptotic stability, only the analytical continuation yielded one ormore complex conjugate pairs of eigenvalues with negative real parts.

Figure 2 shows a plot of the individual eigenvalues as functions of D. Allcurves together form a quartic equation rotated by π/2, and the shape ofthe forth-order polynomial determines the bifurcation pattern. At increasingnegative values D < 0, i.e. in the negative D-direction in figure 2, the twoeigenvalues ε2 and ε3 approach each other and, at some point, D = D1

this pair of real eigenvalues merges and becomes a complex conjugate pair ofeigenvalues. The largest and the smallest eigenvalue, ε1 and ε4, remain single-valued. Because of the shape of a quartic equation, the largest eigenvalue ε1

increases and the lowest eigenvalue ε4 decreases in the negative D-direction.The condition ε1 = 0 occurs at the position D = DoneD, which is defined by

DoneD = − dQ

1 · dQ

2 · d P

1 · d P

2 . (23)

Here, the fixed point P(p1(D), p2(D)

)changes stability and becomes unstable

for D < DoneD. Since only one eigenvalue is involved, the correspondingbifurcation is one-dimensional, for example a transcritical, a saddle-node ora pitchfork bifurcation (For examples see section 4). From equation (23)follows the condition for the stability of fixed points with negative D:

P with Γ(p1, p2) > 0 is stable iff ϑ1 · ϑ2 · Γ(p1, p2) < 1 . (24)

The stability of fixed points P with negative values of D, like their positions(p1, p2), is determined by the two parameter combinations ϑ1, ϑ2, and the

5Generalization to n genes is straightforward: We have 2n variables and 2nfactors rather than four, and the function Γ depends on n protein concentrations.

13

derivatives of the binding functions (22). For D < DoneD the fixed point isunstable, and the largest eigenvalue ε1 is real and positive.

In the direction of positive values, D > 0, the eigenvalues approach eachother in pairs: (ε1, ε2) and (ε3, ε4). If the two eigenvalues in such a pairbecome equal at some value D > 0, at the values D2 and D3 in figure 2, thetwo negative real eigenvalues merge and give birth to a complex conjugatepair with negative real part. The real parts of the two complex conjugatepairs behave like the upper part of a quadratic equation rotated by π/2and hence the real part of the pair formed by the two larger eigenvalues,λ1 = <(ε1, ε2), increases with increasing D. As indicated in figure 2 it maycross zero at some point D = DHopf . There the fixed point looses stabilitythere through a Hopf bifurcation. The value of D can be computed (see theAppendix) and one obtains:

DHopf =(dQ

1 + dQ

2 )(dQ

1 + d P

1 )(dQ

1 + d P

2 )(dQ

2 + d P

1 )(dQ

2 + d P

2 )(d P

1 + d P

2 )

(dQ

1 + dQ

2 + d P

1 + d P

2 )2. (25)

If 0 ≤ D < DHopf is fulfilled for some fixed point P ∈ P with positive D, thefixed point is stable:

P with Γ(p1, p2) < 0 is stable iff − kQ

1 · kQ

2 · kP

1 · kP

2 · Γ(p1, p2) < DHopf . (26)

P is unstable for D > DHopf , and at D = DHopf we expect a marginally stablepoint with concentric orbits in a (small) neighborhood of P . In summary, allfixed points P ∈ P are asymptotically stable in the range DoneD < D < DHopf

(see the Appendix), all four eigenvalues are real between D1 < D < D2.Equation (21) can be solved easily if all degradation rate parameters are

equal, dQ

1 = dQ

n = d P

1 = d P

2 = d:

(ε + d)4 + D = 0 =⇒ εi = − d + 4√−D , i = 1, . . . , 4 .

Similarly, the eigenvalues are readily calculated if all RNA and all proteindegradation rates are the same: dQ

1 = dQ

2 = dQ and d P

1 = d P

2 = d P yields

(ε + dQ)(ε + d P) = ±√−D ,

where the computation boils down to solving two quadratic equations.For a given fixed point the function Γ(p1, p2) determines the bifurcation

behavior of the system. In all examples with simple binding functions oftype (5), Γ(p1, p2) is either positive or negative for all (non-negative) valuesof the concentrations p1 and p2. Indeed we find Γ(p1, p2) ≥ 0 for activationof both genes (act-act) and repression of both genes (rep-rep), whereas com-binations of activation and repression, (act-rep) and (rep-act), yields alwaysnon-positive values, Γ(p1, p2) ≤ 0. According to (21) D has opposite sign toΓ. Calculation of the regulatory determinant for arbitrary n is straightfor-ward and yields:

D = ∓ kQ

1 kQ

2 kP

1kP

2

n2K1K2 pn−11 pn−1

2

(K1 + pn1 )2(K2 + pn

2 )2. (27)

Here, the minus sign holds for act-act and rep-rep whereas the plus sign istrue for act-rep and rep-act. In case of act-act insertion of the coordinates of

14

the fixed point at the origin, P1 = (0, 0), yields the very general result thatP1 is always stable for n ≥ 2, because we obtain D = 0 in this case.

Equations (17-19) are useful in searching parameter space for bifurca-tions. Auxiliary variables can be used to define manifolds on which thesearch is carried out. As an illustrative example we consider the search for aHopf bifurcation along the one-dimensional manifold defined by (kQ

1 = χ1 · s,kQ

2 = χ2 · s,K1 = λ1/s,K2 = λ2/s) in the act-rep system (18). From theserelations follows ϑi = δi ·s with δ1 =

(kP

1/(dQ

1 d P

1 ))χ1 and δ2 =

(kP

2/(dQ

2 d P

2 ))χ2,

respectively. The computation of the equilibrium concentrations for large sis straightforward and yields for n > 1:

p1 = α1 · s2/(n2+1) with α1 =

(δ1(δ2λ1)

n

λ2

)1/(n2+1)

and

p2 = α2 · s−2n/(n2+1) with α2 =

(λ1

(δ1(δ2λ1)n3/(n2+1)

)n/(n2+1)

.

Insertion into the expression for the regulatory determinant leads to exactcancellation of the powers of s and we find in the

limit of large s : Dlim ≈ kQ

1 kQ

2 kP

1kP

2

n2

ϑ1ϑ2

= dQ

1 dQ

2 d P

1 d P

2 n2 . (28)

This value has to be compared with the condition for the occurrence of a Hopfbifurcation (25). As an example of an application we analyze the function

H(dQ

1 , dQ

2 , d P

1 , d P

2 , n) = Dlim

/DHopf =

= n2 dQ

1 dQ

2 d P

1 d P

2 (dQ

1 + dQ

2 + d P

1 + d P

2 )2

(dQ

1 + dQ

2 )(dQ

1 + d P

1 )(dQ

1 + d P

2 )(dQ

2 + d P

1 )(dQ

2 + d P

2 )(d P

1 + d P

2 ), (29)

to show whether or not act-rep systems with Hill coefficient n > 1 can un-dergo a Hopf bifurcation at certain parameter values and sustain undampedoscillations. A value H > 1 indicates that a limit cycle exists for sufficientlylarge values of s. The maximum of H is computed by partial differentiationwith respect to the degradation rate constants6

(∂H

∂dQ

1

)= 0 =⇒ (dQ

1 )3(dQ

2 + d P

1 + d P

2 ) + (dQ

1 )2((dQ

2 )2 + (d P

1 )2 + (d P

2 )2)−

− 3dQ

1 (dQ

2 d P

1 d P

2 ) − dQ

2 d P

1 d P

2 (dQ

2 + d P

1 + d P

2 ) = 0 .

This cubic equation is hard to analyze but the question raised here can beanswered without explicit solution. We assume dQ

2 = d P

1 = d P

2 = d and obtain

(dQ

1 − d)(dQ

1 + d)2 = 0 =⇒ dQ

1 = d and H(d, d, d, d, n) =n2

4.

By numerical inspection we showed that any deviation from uniform degra-dation rate parameters leads to a smaller value for the maximum of H. In

6Since the function (29) is symmetric with respect to all four rate parametersall four partial derivatives have identical analytical expressions.

15

the strong binding limit the act-rep system with n = 2 is confined to valuesH ≤ 1 and indeed no limit cycle has been observed. Systems with n ≥ 3,however, show values of Hmax = n2/4 > 1 in certain regions of parameterspace, and they do indeed sustain undamped oscillations. For the fixed pointin the positive quadrant the D-value increases from weak to strong binding(See section 4.2) and this completes the arguments for the nonexistence ofundamped oscillation for n = 2.

3.3 Basal transcription

The basal state shown in figure 1 is often characterized as ‘leaky transcrip-tion’ since it leads to low levels of mRNA. In order to take basal activityformally into account we add (small) constant terms, γ1 and γ2, to the bind-ing functions (5) and find for activation and repression

F(act)1 (p2) = γ1 +

pn2

K2 + pn2

=γ1K2 + (1 + γ1)p

n2

K2 + pn2

F(rep)1 (p2) = γ1 +

K2

K2 + pn2

=(1 + γ1)K2 + γ1p

n2

K2 + pn2

F(act)2 (p1) = γ2 +

pn1

K1 + pn1

=γ2K1 + (1 + γ2)p

n1

K1 + pn1

F(rep)2 (p1) = γ2 +

K1

K1 + pn1

=(1 + γ2)K1 + γ2p

n1

K1 + pn2

.

(30)

Basal transcription activity is readily incorporated into the analytic pro-cedure described here. The computation of fixed points is straightforwardalthough it involves more terms. Since the constant terms vanish throughdifferentiation, the regulatory determinant and the whole Jacobian matrixdepend on basal transition only via the changes in the positions of the fixedpoints, P = (p1, p2).

For gene regulation leaky transcription is most important in cases whereboth genes are activated. Activation without basal transcription allows forirreversible silencing of both genes since they can be turned off completelyand after degradation of the activator proteins the system cannot recoverits activity. In mathematical terms the origin, P (0, 0), is an asymptoticallystable fixed point. Basal transcription changes this situation because somelow-level protein synthesis is always going on and the origin is a fixed pointno longer. In the forthcoming section we shall consider several exampleswhere leaky transcription has been included.

16

4 Selected examples

Examples for activation and repression were considered for non-cooperativebinding (n = 1) as well as for cooperative binding (n ≥ 2) up to n = 4. Inaddition, examples were included where intermediate complexes are active intranscription. Our calculations have shown that at low ratios ϑ/K all sys-tems sustain asymptotically stable stationary states in the positive quadrantincluding the origin, all except very few (see table 4) undergo a bifurcation atsome larger value of ϑ/K and reach, thereafter, the relevant or regulated state.The changes in the dynamical patterns in parameter space are investigatedby means of an auxiliary variable s that defines a path in parameter space(See section 3.2). The range in parameter space with low values ϑ/K is char-acterized by low ratios of reaction rate parameters and/or high dissociationparameters of the regulatory complexes, which is tantamount to low bindingconstants or low affinities. It will be denoted here as the unregulated regime,because the dynamics in this range is not suitable for regulatory functions.In contrast, the parameter range with high ratios ϑ/K above the bifurcationvalue will be called the regulated regime since bistabilities or oscillations (orsometimes both) occur in this region. In the cases discussed here we shallinvestigate paths through parameter space that lead from the unregulated tothe regulated regime which can be achieved, for example, by assuming ϑ ∝ sand K ∝ s−1.7 For all pure activation-activation and repression-repressionsystems the function D is non-positive and hence Hopf bifurcations, and limitcycles derived from them, can be excluded. Instead one dimensional bifur-cations, transcritical, saddle-node, and pitchfork bifurcations, are observed,the latter two resulting in bistability of the system. Activation-repressionyields non-negative values of D and hence the systems may reach oscillatorystates via the Hopf bifurcation mechanism. Examples, where intermediatecomplexes are active in transcription, were included because they may giverise to regulatory determinants D that can adopt positive as well as nega-tive values and therefore may sustain oscillations and bistability at differentparameter values.

4.1 The non-cooperative binding case

In the non-cooperative case binding of effectors to the regulatory regions ofDNA is described by the equilibria (3) and (4). As said we have two classesof binding functions (5) and this leads to three different cases: (i) activation-activation, (ii) activation-repression, and (iii) repression-repression, whichwill be handled separately.

Activation-activation. The binding functions for this case are

F1(p2) =p2

K2 + p2

and F2(p1) =p1

K1 + p1

. (31)

7Considering the limits, lims→0 Pk(s) and lims→∞ Pk(s) with k = 1, 2, . . . , isimportant for all fixed points, for example, in order to recognize equivalent andnon-equivalent paths through parameter space.

17

Figure 3: Position and stability of the two fixed points in the two-gene

non-cooperative activation-activation system according to equation (32).The upper part of the figure shows the regulatory determinant D as a function ofthe auxiliary variable s for both fixed points P1 and P2. According to equation (23)we observe a transcritical bifurcation at the value s = soneD = 0.559. Both D-functions adopt the value DoneD = −1 and exchange stability at this point. Theorigin, P1, is asymptotically stable for s < soneD whereas P2 shows stability abovethis value. The lower plot shows the position of the two fixed points as a functionof s. Parameter values: kQ

1 = kQ

2 = 1, K1 = 0.5/s, K2 = 2.5/s, kP

1 = kP

2 = 2, anddQ

1 = dQ

2 = d P

1 = d P

2 = 1. Color code see ‘Notation’.

The search for stationary points leads to two solutions:

P1 = (0, 0) and P2 =

(ϑ1ϑ2 − K1K2

ϑ2 + K2

,ϑ1ϑ2 − K1K2

ϑ1 + K1

). (32)

For ϑ1ϑ2 > K1K2 the fixed point P2 is inside the positive quadrant of proteinspace and it is stable as can be readily verified by means of equation (24).At the critical value ϑ1ϑ2 = K1K2 the two fixed points exchange stability asrequired for a transcritical bifurcation. A special example is shown in figure 3:The path through parameter space is defined by K1 = 0.5/s and K2 = 2.5/s(The other parameters are summarized in the caption of figure 3). The limitsfor the position of the two fixed points are: (i) P1 stays at the origin for all s,and (ii) for P2 we compute lims→0 P2 = (−∞,−∞) and lims→∞ P2 = (2, 2).

18

Figure 4: Position and stability of the two fixed points in the two-gene

non-cooperative activation-repression system. The upper part of the figureshows the regulatory determinant D as a function of the auxiliary variable s forboth fixed points P1 and P2. The lower plot shows the position of the two fixedpoints as a function of s. The ’non-biological’ fixed point P1 lies in the negativequadrant, the regulatory determinant D(P1) adopts the value D = 4 at s = 2 andthe system undergoes a Hopf bifurcation there (figure 5). The relevant fixed pointP2 is asymptotically stable for all values of s. Parameter values: kQ

1 = kQ

2 = 1,K1 = K2 = 1/s, kP

1 = kP

2 = 2, and dQ

1 = dQ

2 = d P

1 = d P

2 = 1. Color code see‘Notation’.

It is worth considering the physical meaning of the stability condition forP2: Since the parameters ϑ are the squares of the geometric means of theformation rate constants divided by the degradation rate constants and theK’s are the reciprocal binding constant, both genes are active for sufficientlylarge formation rate parameters and high binding affinities. The combinationactivation-activation with non-cooperative binding shows modest regulatoryproperties. It sustains two states: (i) A regulated state where both genes aretranscribed and (ii) a state of ’extinction’ with both genes silenced.

19

Figure 5: A Hopf bifurcation at P2 in the two-gene non-cooperative

activation-repression system. From the dependence of the regulatory deter-minant D at P1 on the auxiliary variable s shown in the upper plot of figure 4a bifurcation with D = 4 is expected to occur at the value s = 2. We show tra-jectories for s = 1 (upper plot; all parameter values identical with the choice infigure 4) and s = 4 (lower plot) and observe spiralling out and spiralling in, respec-tively. Color code: The projection of the trajectory onto the mRNA concentrationsubspace,

(q1(t), q2(t)

), is shown in red and the projection onto protein subspace,(

p1(t), p2(t))

is plotted in blue.

Activation-repression For activation of gene one and repression of genetwo the binding functions are of form:

F1(p2) =p2

K2 + p2

and F2(p1) =K1

K1 + p1

. (33)

Here the stationary conditions sustain two fixed points that are obtained assolutions of the quadratic equation

K2 p21 + (ϑ2 + K2) K1 p1 − ϑ1 ϑ2 K1 = 0 .

This equation has one positive and one negative solution for p1. For knownp1 the second protein concentration is calculated from

p2 =ϑ2 K1

K1 + p1

.

20

Combining this equation with p1 = ϑ1p2/(K2 + p2) allows to prove thatboth variables, p1 and p2, have the same sign and accordingly, one fixedpoint, P1 lies inside the negative quadrant of the (p1, p2)-space and plays norole in biology. The second stationary point is characterized by two positiveconcentration values, lies inside the positive quadrant, and is stable (figure 4).

The properties of the fixed point P1 are, nevertheless, useful for the analy-sis of the dynamical system in the sense of continuation into the neighboringquadrants. In particular,it allows for an inspection of the condition (26).Differentiation shows that the function D(p1, p2) of equation (21) is alwayspositive and then the observation of a Hopf bifurcation cannot be excluded.In the example shown in figure 4 D(P1) adopts indeed the value D = 4 atP1 (for the auxiliary parameter s = 2). As shown in figure 5 the trajectorieschange from spiralling in at s = 4 to spiralling out at s = 1. For carefulparameter choices a small limit cycle is observed further apart from the (un-stable) fixed point the trajectories of the dynamical system diverges whenthey come close to the lines p1 = −K1 and p2 = −K2.

In summary we obtain only the scenario of the unregulated regime andno bifurcation to a state that is interesting from the point of regulation.The system is characterized by a single state, which is stable for all physicalparameter values, and shows no potential regulatory properties.

Repression-repression. Both genes code for repressors in the third sce-nario. The binding functions are of the form

F1(p2) =K2

K2 + p2

and F2(p1) =K1

K1 + p1

, (34)

and the two stationary solutions are again obtained as solutions of a quadraticequation,

K2 p21 + (ϑ2K1 − ϑ1K2 + K1K2) p1 − ϑ1 K1 K2 = 0 ,

and the same relation as in the previous section

p2 =ϑ2 K1

K1 + p1

.

Similar as in the previous example it can be shown that p1 and p2 have alwaysthe same sign. One of the two solutions is unstable and lies in the negativequadrant whereas the other one, the physically meaningful solution, is situ-ated in the positive quadrant and it is asymptotically stable (figure 6). Theobserved stabilities are readily predicted from inspection of equation (21):Since D is non-positive we have either four real negative eigenvalues or tworeal eigenvalues and a complex conjugate pair with a real part lying betweenthe other two (figure 2). The stable fixed point is identified by 0 ≥ D ≥ −1,the unstable one by D < −1.

In the non-cooperative binding case the combination repression-repressiongives rise to a single state only, it represents the unregulated scenario, andit is not suitable for regulation.

21

Figure 6: Position and stability of the two fixed points in the two-

gene non-cooperative repression-repression system. As in the activation-activation case the regulatory determinant is non-positive, D ≤ 0. The topmostplot shows D(p1, p2) as a function of the auxiliary variable s for both fixed points.The fixed point P1 lies in the negative quadrant and is unstable, D < −1. The fixedpoint P2 is always situated in the physical protein space, the positive quadrant,and it is asymptotically stable since 0 ≥ D ≥ −1 is fulfilled. Choice of parameters:kQ

1 = kQ

2 = 2 · s, K1 = K2 = 1, and kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. Color codesee ‘Notation’.

Activation with leaky transcription. The effect of leaky transcriptionis illustrated in figures 7 and 8. For γi > 0 (i = 1, 2) the fixed point at theorigin is shifted either into the negative or into the positive quadrant suchthat exactly one fixed point is in each quadrant. The fixed point in physicalprotein space is always asymptotically stable, the one outside physical spaceis unstable. The scenario shown in figure 7 starts out from the position of thetranscritical bifurcation in the limit γ → 0. Accordingly, two states fulfillingthe criteria mentioned above emerge at γ > 0, and the unstable state appearsin the negative quadrant, P1 = (p

(1)1 < 0, p

(1)2 < 0) and the stable fixed point

is always inside the positive quadrant, P2 = (p(2)1 > 0, p

(2)2 > 0).

The plots in figure 8 illustrate the influence of weak basal transcriptionon the activation-activation regulatory system around the transcritical bifur-cation point of the unperturbed system. An auxiliary variable is defined by

22

Figure 7: Position and stability of the two fixed points in the two-gene

non-cooperative activation-activation system with leaky transcription:

γ-dependence. Leaky transcription is introduced into the system exactly at thetranscritical bifurcation point. The upper plot shows D(p1, p2) for both fixedpoints as a function of an auxiliary variable s, which measures the extent of basaltranscription, γ1 = γ2 = 0.001 · s. The lower plot presents the positions of thetwo fixed points. For s > 0 the fixed point P1 lies in the negative quadrant and isunstable, D < −1. The fixed point P2 is always situated in the physical proteinspace, the positive quadrant, and it is asymptotically stable since 0 ≥ D ≥ −1is fulfilled. Choice of parameters: kQ

1 = kQ

2 = 1, K1 = 0.892857, K2 = 4.464286,kP

1 = kP

2 = 1, dQ

1 = dQ

2 = 2, and d P

1 = d P

2 = 1. Color code see ‘Notation’.

K1 = 0.5/s and K2 = 2.5/s (The other parameter values are given in thecaption of figure 8) and transcritical bifurcation is observe the at s = 0.56(γ1 = γ2 = 0). Small gamma values gives rise to avoided crossing : At somedistance from the virtual crossing point the two states are very close to thoseof the pure activation system and the continuation of these states at theother side of the virtual bifurcation point is readily recognized. The splittingfor increasing γ ≥ 0 at exactly this point was shown in the previous figure 7.

4.2 The cooperative binding case

For integer Hill coefficients n ≥ 2 the binding curves have sigmoidal shape,the dynamics of the systems becomes richer and multiple steady states or

23

Figure 8: Position and stability of the two fixed points in the two-gene

non-cooperative activation-activation system with leaky transcription:

K-dependence. In contrast to figure 7 basal transcription occurs at a constantrate γ1 = γ2 = 0.001 as a function of the equilibrium parameters, K1 = 0.5/s andK2 = 2.5/s. The upper plot shows D(p1, p2) for both fixed points as a functionof an auxiliary variable s in the neighborhood of the transcritical bifurcation ats = 0.56 for γ1 = γ2 = 0. The lower plot presents the positions of both points inthe range of s. For s > 0 the fixed point P1 lies in the negative quadrant and isunstable, D < −1. The fixed point P2 is always situated in the physical proteinspace, the positive quadrant, and it is asymptotically stable since 0 ≥ D ≥ −1 isfulfilled. Choice of other parameters: kQ

1 = kQ

2 = 1, kP

1 = kP

2 = 1, dQ

1 = dQ

2 = 2,and d P

1 = d P

2 = 1. Color code see ‘Notation’.

oscillatory behavior emerge. As mentioned already in section 3.1 the poly-nomials for the computation of the positions of fixed points have an odd oreven number of real solution for even or odd Hill coefficients n. Accordinglywe find two or four solutions for n = 1 and n = 3, and one or three solu-tions for n = 2 and n = 4, respectively. Despite the high degrees of thepolynomials in p1 or p2 (n2 + 1) we did not detect more stationary states upto n = 4. Although our searches of the high-dimensional parameter spaceswere not exhaustive, it is unlikely that fixed points remained unnoticed. Therather small number of distinct states causes the dynamic patterns of thecooperative systems with different n ≥ 2 to be qualitatively similar, with theonly exception being activation-repression with n = 2, and therefore we shallgroup them only according to activation, repression, and basal transcription.

24

Figure 9: Position and stability of the fixed points in the two-gene

cooperative activation-activation system with Hill coefficient n = 2. Theproperties of the system are studied as a function of the auxiliary variable s,which defines the binding constants: K1 = k2 = 0.5/s. The systems shows asaddle-node bifurcation at s = 0.5. Choice of the other parameters: kQ

1 = kQ

2 = 2,kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. Color code see ‘Notation’.

Activation-activation. For n = 2 the expansion of equation (16) yields apolynomial of degree five. Numerical solution yields one or three solutions inthe positive quadrant including the origin which correspond to one or threesteady states. The origin represents one fixed point, P1 = (0, 0) that, incontrast to the non-cooperative system is always stable.8 Searching parameterspace in the direction of increasing transcription rate parameters, (kQ

1 =χ1 · s, kQ

2 = χ2 · s), and/or decreasing dissociation constants of regulatorycomplexes, (K1 = λ1/s,K2 = λ2/s), yields a saddle-node bifurcation whenthe condition D = DoneD of equation (23) is fulfilled (figure 9). At thispoint, separating the unregulated regime with the origin being the only stablestate from the regulated regime, two new fixed points appear and branch off,thereby fulfilling the conditions D < DoneD and D > DoneD, respectively. Theformer fixed point is unstable – at least for some range in parameter space –whereas the latter fixed point is asymptotically stable since D can only adoptnegative signs (For examples with no sign restriction on D see below in the

8This result follows straightforwardly from a computation of the derivatives inthe Jacobian, which yields D = 0 at the origin for all Hill coefficients n > 1.

25

Table 2: Dependence of the bifurcation point on the Hill coefficient n.

The value of the auxiliary variable s at the bifurcation point that separates theunregulated regime and the regulated regime is compared for different cooperativeregulation modes and Hill coefficients n = 2, 3, 4. In order to allow for compari-son equivalent paths through parameter space were chosen for all three classes ofsystems.

System Bifurcation Parameter Variable s at bifurcation

type variation n = 2 n = 3 n = 4

act-act a saddle-node K1 = K2 = 0.5/s b 0.5 0.422 0.296

act-rep Hopf K1 = K2 = 0.5/s b – 2.772 0.5

rep-rep pitchfork c K1 = K2 = 0.5/s b 0.5 0.106 0.033

a In case of act-act other paths through parameter space lead to small or almost vanishingdependencies of the bifurcation value of s on the Hill coefficient n, for example we founds = 0.79, 0.81, 0.78 for n = 2, 3, 4, kQ

1 = kQ

2 = 2 · s, K1 = K2 = 0.5/s and s = 1, 0.96, 0.90for kQ

1 = kQ

2 = 2 · s, K1 = K2 = 1/s, respectively (All other parameters being one).

b The other parameter values were: kQ

1 = kQ

2 = 2 and kP1 = kP

2 = dQ

1 = dQ

2 = d P1 = d P

2 = 1.

c The pitchfork bifurcation becomes a saddle-node bifurcation, when the symmetry con-sisting of identical parameters for gene 1 and gene 2 is broken (See figure 13).

paragraph dealing with intermediate cases). Raising the Hill coefficient fromn = 2 to n = 3 and to n = 4 has rather little effect on the position of thebifurcation point. As shown in table 2 we find somewhat smaller values of sat the bifurcation point for the higher Hill coefficients, but the changes aremuch smaller than for the activation-repression and the repression-repressionsystem. In addition this weak dependence is replaced by even weaker or nodependence on n when the implementation of the auxiliary variable s ischanged.

Activation-repression. The activation-repression with n = 2 is charac-terized by a nonnegative regulatory determinant (D ≥ 0), but as discussed insection 3.2 the maximal value of D is insufficient for a Hopf bifurcation. Thesystem exhibits only one stable fixed point and no undamped oscillations canoccur. In other words, the act-rep systems with n = 1 and n = 2 show onlyan ’unregulated regime’.

For n ≥ 3, however, a Hopf bifurcation is predicted and a limit cycle canbe observed for sufficiently strong binding (figures 10 and 11). Systems of thisclass exhibit periodically changing gene activities. Oscillation in regulatorysystems can be used as a pacemaker inducing periodicity into metabolismas it occurs, for example, in circadian and other rhythms. The qualitativepicture of the bifurcation diagram is essentially the same for Hill coefficientsn = 4 and larger. Along equivalent trajectories leading from the unregulatedto the regulated regime the Hopf bifurcation occurs at substantially smallers-values than for n = 3. In other words, the regulated domain in parameterspace – here the domain that contains an unstable fixed point and a limit

26

Figure 10: Position and stability of the fixed points in the two-gene

cooperative activation-repression system with Hill coefficient n = 3. Theregulatory determinant is non-negative, D ≥ 0. The properties of the dynamicalsystem, shown here as functions of the auxiliary variable s, kQ

1 = kQ

2 = 2s and K1 =K2 = 0.5/s, change at the Hopf bifurcation observed at s = 1.2903 (vertical linein the plots): The central fixed point becomes unstable and a limit cycle appears(See figure 11). Choice of other parameters: kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1.Color code see ‘Notation’.

cycle – becomes larger with increasing cooperativity as expressed by higherHill coefficients.

Repression-repression. The cooperative repression-repression system isthe prototype of a genetic switch. At low affinities the system sustains oneasymptotically stable stationary state. In the regulated regime it shows bista-bility consisting of two asymptotically stable states that can be characterizedas G1 active and G2 silenced and vice versa, G2 active and G1 unregulated.The two states are separated by a saddle point (figures 12 and 13). Forsymmetric choices of parameters, implying that all parameters for G1 havevalues identical to those of the corresponding parameters for G2, a pitchforkbifurcation separates the unregulated regime from the regulated regime. In-troducing asymmetry through different values for kQ

1 and kQ

2 and/or K1 andK2, respectively, removes the degeneracy leading to the pitchfork and a sad-

27

Figure 11: Stable fixed point and limit cycle in the two-gene cooper-

ative activation-repression system with Hill coefficient n = 3. The twoplots show trajectories of the system before and after the Hopf bifurcation thatoccurs at D(s) = 4 with sHopf = 2.17 in the system with kQ

1 = kQ

2 = 1 · s andK1 = K2 = 0.5/s and kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. The upper plot shows atrajectory for s = 2 (D(2) = 3.6804) with a stable fixed point and the trajectoriesspiralling inwards, the lower plot was recorded for s = 2.5 (D(2.5) = 4.5265) wherethe fixed point is unstable and a stable limit cycle is observed. The trajectoriesspiral form outside towards the limit cycle. For initial conditions near the unstablefixed point the limit cycle is approached through spiralling outwards (not shown).Color code: The projection of the trajectory onto the mRNA concentration sub-space,

(q1(t), q2(t)

), is shown in red and the alternative projection onto the protein

subspace,(p1(t), p2(t)

)is plotted in blue.

dle node bifurcation remains. As illustrated nicely by the parametric plotin the middle of figure 13 the stable fixed point of the unregulated regimeis attracted towards the (no more existing) point of the pitchfork. Such aphenomenon is often called the influence of a ‘ghost’ on bifurcation lines ortrajectories. Considering identical paths through parameter space the po-sition of the pitchfork bifurcation is shifted towards smaller values of theauxiliary parameter s in the series n = 2, 3, 4 (table 2).

28

Figure 12: Position and stability of the two fixed points in the two-

gene cooperative repression-repression system with n = 2 and symmetric

choice of parameters. In the symmetric case the one dimensional bifurcationis a pitchfork at the value s = 0.7937 for the following choice of parameters:kQ

1 = kQ

2 = 2 · s, K1 = K2 = 0.5/s, and kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1.The topmost plot shows the dependence of the regulatory determinants on s (Bysymmetry reasons the violet and the turquoise curve are on top of each other andonly the latter can be seen). in the middle we present a parametric plot of thepositions of all fixed points,

(p1(s), p2(s)

), and the picture at the bottom shows

these positions as function of s. Color code see ‘Notation’.

29

Figure 13: Position and stability of the two fixed points in the two-

gene cooperative repression-repression system with n = 2 and asym-

metric choice of parameters. In the asymmetric case the pitchfork bifurcationis replaced by a saddle node bifurcation that occurs here at s = 1.1515 for theparameter choice: kQ

1 = 1.9 · s, kQ

2 = 2 · s, K1 = 0.55/s, K2 = 0.45/s, andkP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. The topmost plot shows the dependence ofthe regulatory determinants on s, in the middle we present a parametric plot of thepositions of all fixed points,

(p1(s), p2(s)

), and the picture at the bottom shows

the coordinates of the fixed points as functions of s. Color code see ‘Notation’.

30

Figure 14: Position of fixed points in the two-gene cooperative

repression-repression system. The parametric plot shows a superposition ofthe pitchfork diagrams for the Hill coefficients n = 2, 3, and 4 with the varied pa-rameters kQ

1 = kQ

2 = 2·s and K1 = K2 = 0.5/s defining the path through parameterspace. Choice of the other parameters: kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. Colorcode see ‘Notation’.

Genetic switches with different Hill coefficients. The regulatoryproperties of repression-repression systems are of primary interest in compu-tations of genabolic networks. As said above, the value of s at the pitchforkor saddle-node bifurcation decreases substantially for increasing Hill coeffi-cients (table 2). Two more properties are highly relevant in the context ofregulation: (i) The ‘pitchforks’ in parametric plots for different Hill coef-ficients are surprising similar (figure 14) and (ii) the regulatory selectivityincreases strongly in the sequence n = 2, 3, and 4 (table 3).

The superposition of the three pitchfork diagrams in figure 14 revealsastonishing agreements of the plots for three different Hill coefficients (n =2, 3, 4). This general behavior is changed slightly only when different pathsthrough parameter space are chosen as long as kQ

1 = χ1 · s and kQ

2 = χ2 · sis applied for the kinetic parameters. Constant values of K1 and K2, forexample, have little influence on the diagram. If the dissociation constants,however, are varied, for example K1 = λ1/s and K2 = λ2/s, and two kineticparameters are chosen to be constant, the bifurcation diagram changes shapesubstantially. The differences in the plots are explained readily by inspectionof the limits derived for the paths through parameter space. As an examplewe present the limits of the fixed points for the two cases mentioned above(See also table 1):

ϑ1 = δ1 · s, ϑ2 = δ2 · s,K1, K2 : lims→0

P1 = (0, 0) , lims→∞

P1 = (∞,∞)

lims→∞

P2 = (∞, 0) , lims→∞

P3 = (0,∞)

ϑ1, ϑ2, K1 = λ1/s,K2 = λ2/s : lims→0

P1 = (ϑ1, ϑ2) , lims→∞

P1 = (0, 0)

lims→∞

P2 = (ϑ1, 0) , lims→∞

P3 = (0, ϑ2) .

Simultaneous variation of the ϑ-parameters and the equilibrium constants

31

Table 3: Position of the bifurcation point in repression-repression sys-

tems and switching efficiency for different Hill coefficients n = 2, 3, 4.The values in the table are sampled on equivalent paths through parameter spacewith the following parameter values: kQ

1 = kQ

2 = 2, K1 = K2 = 1/s andkP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. Because of symmetry the bifurcation isof pitchfork type.

Variable s at Position Silencing efficiency a

n bifurcation p1 = p2 s = 1.5 s = 2.5 s = 4.0

2 1 1 (1.577,0.423) (1.775,0.225) (1.866,0.134)

3 0.2109 1.333 (1.989,0.156) (1.996,0.096) (1.998,0.061)

4 0.0685 1.500 (2.000,0.080) (2.000,0.049) (2.000,0.031)

a The values in parentheses represent the stationary concentrations of regulators proteins,(p1, p2), at the fixed point P1 for the given value of the auxiliary variable s.

K results in the same behavior as variation of the former parameters alone.Clearly, only patterns with the same limits are comparable and in the currentexample we chose the former case, variation of kinetic parameters with orwithout variation of dissociation constants.

In table 3 the efficiency of genetic switches is compared for different Hillcoefficients. The numbers illustrate the effect of higher order cooperativity:The higher the value of n, the larger is the selective power of the switch.At s = 4, for example, we find the mole fractions9 x2 = 0.067, 0.030, and0.015 for the protein of the silenced gene for the Hill coefficients n = 2, 3,and 4, respectively. Since the pitchfork diagrams are not very different forthe three cases the efficiency in silencing is caused by the different values ofs at comparable points. An illustration for this argument is given by thebifurcation point itself which occurs at s = 1, 0.211, and 0.069 for n = 2, 3,and 4, respectively.

Summarizing cooperative repression-repression systems we recognize theirimportance as genetic switches. Hill coefficients higher than n = 2 have twoproperties that are relevant for regulation: (i) The regulated regime comprisesa larger domain in parameter space, and (ii) the selectivity of the regulatoryfunction increases with increasing n.

Activation with basal transcription. The cooperative case of activation(n=2) with leaky transcription provides an illustrative example of a systemwith two saddle-node bifurcations, (soneD)1 and (soneD)2, which gives rise tohysteresis. Figure 15 presents the fixed points as functions of spontaneoustranscription rate parameter γ. In this figure the parameters were chosensuch that the system has only one fixed point at lim γ → 0, the stableorigin. With increasing values of γ the system undergoes a saddle-nodebifurcation that leads to the regulated regime with two asymptotically stablefixed points, one at high and one at low stationary protein concentrations,

9The mole fraction is defined by xi = pi/(p1 + p2) for i = 1, 2.

32

Figure 15: Position and stability of the two fixed points in the two-

gene cooperative activation-activation system with leaky transcription.

The influence of basal transcription activity on the bifurcation behavior of theact-act system with n = 2 is illustrated by variation of γ as in figure 7 accordingto γ1 = γ2 = 0.001 · s. The system passes through two saddle node bifurcationsat s = 23.81 and s = 74.92, and it shows hysteresis. Choice of parameters:kQ

1 = kQ

2 = 2, K1 = K2 = 1.1, and kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1. Color codesee ‘Notation’.

separated by a saddle. Further increase in γ, however, leads to a secondsaddle-node bifurcation that annihilates the stable state originating from theorigin together with the unstable saddle. The state that eventually remainsis the high activity state (both genes active) which originates in the firstsaddle-node bifurcation.

As shown in figure 15 the sequence of bifurcations gives rise to hysteresisin the range between the two bifurcation points, (soneD)1 < s < (soneD)2.Coming from high values of γ the system stays in the high protein concen-tration branch (as long as it is not shifted to the low concentration state byfluctuations), the low protein concentration branch, on the other hand, isreached from s values below (soneD)1.

The existence of a single stable state at high values of s and with highprotein concentrations is easy to predict from the reaction kinetics: Withincreasing γ-values spontaneous transcription will, at some point, dominateand then only the non-regulated stationary state exists. This situation, how-

33

ever, is unlikely to occur in realistic biological systems, because unregulatedtranscription is common at very low levels only. The second saddle-nodebifurcation – although not natural – could well be of interest for the designof artificial regulatory systems since it allows for up and down regulation ofgene activity in an intermediate range.

Influence of basal transcription on bifurcation patterns. The influ-ence of basal transcription on all three classes of regulatory systems (act-act,act-rep, and rep-rep) is compared in figure 16. For simplicity the diagramsshow only the symmetric cases, γ1 = γ2 = γ and ϑ1 = ϑ2 = ϑ, and the sys-tems with the lowest values of the Hill coefficient at which the characteristicbifurcation pattern appears (n = 2 for act-act and rep-rep, and n = 3 foract-rep). All three systems have in common that the bifurcations vanish atsufficiently large values of γ > γcrit and then the systems sustain only onestable state. Below γcrit we find the specific bifurcation pattern for the sys-tems in a certain range ϑ

(1)crit < ϑ < ϑ

(2)crit: The bifurcation at low values of ϑ is

compensated by an inverse bifurcation of the same class at higher ϑ-values.For all systems the inverse bifurcation point approaches infinity for vanishingbasal transcription: limγ→0 ϑ

(2)crit = +∞. There is, however, one characteristic

difference between the three bifurcation diagrams: The lower bifurcation linehas a negative slope in the act-act system but a positive slope in the twoother classes, act-rep and rep-rep. This implies that for increasing basaltranscription, γ < γcrit the saddle-node bifurcation occurs at lower valuesof ϑ, whereas increasing basal activity drives the first bifurcation to higherϑ-values in the other two classes of systems.

Two examples of intermediate regulation. In order to illustrate the bi-furcation pattern with respect to regulatory determinants D that can changesign we consider two examples of regulation by intermediate complexes:(i) a combination of activation and intermediate regulation and (ii) a com-bination of repression and intermediate regulation. Both cases have a Hillcoefficient n = 4. The four complexes formed by successive binding are showtogether with the various forms of potential transcriptional regulation in fig-ure 17. The four dissociation constants are multiplied to yield the followingcombinations:

κ11 = K(1)1 · K(1)

2 · K(1)3 · K(1)

4 , κ12 = K(1)2 · K(1)

3 · K(1)4 , κ13 = K

(1)3 · K(1)

4 , κ14 = K(1)4

κ21 = K(2)1 · K(2)

2 · K(2)3 · K(2)

4 , κ22 = K(2)2 · K(2)

3 · K(2)4 , κ23 = K

(2)3 · K(2)

4 , κ24 = K(2)4 .

As mentioned in section 2.1 the equilibrium parameters used here are macro-scopic dissociation constants.

In the first system the saturated complex H(4)2

(F1(p2)

)and the ‘inter-

mediate 2’ H(2)2

(F2(p1)

)initiate transcription and the binding functions are

of the form

F1(p2) =p4

2

κ21 + κ22p2 + κ23p22 + κ24p3

2 + p42

and

F2(p1) =κ13 p2

1

κ11 + κ12p2 + κ13p21 + κ14p3

1 + p41

.

34

Figure 16: The influence of basal transcription on the bifurcation pat-

terns of gene regulation. The topmost plot presents the positions of the saddle-node bifurcations of the act-act system with Hill coefficient n = 2 in the (ϑ, γ)-plane. In the area to the right of the first and below the second curve we observethree stationary states, elsewhere one stable stationary state. The plot in themiddle refers to the act-rep system with Hill coefficient n = 3: Oscillations oc-cur below the bifurcation curve. The plot at the bottom shows the analogouscurve of the rep-rep system with Hill coefficient n = 2. Three steady statesare observed between two (opposite) pitchfork bifurcations, i.e. in the area be-low the curve. Other parameters: K1 = K2 = 0.5, and kQ

1 = ϑ1 = kQ

2 = ϑ2,kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1 for the middle plot.

35

Figure 17: Intermediate complexes as active forms in transcription.

A four step binding equilibrium of four monomers to a binding site in the generegulatory region is shown as an example for the dynamics of regulation by means

of active intermediate complexes. Three intermediate complexes, H(1)i , H

(2)i , and

H(3)i are potential candidates for transcription.

Computation of the regulatory determinant D is straightforward and yields

D(p1, p2) = −kQ

1 kQ

2 kP1kP

2 ·

· κ13p1p32(2κ11 + κ12p1 − κ14p

31 − 2p4

1) (4κ21 + 3κ22p2 + 2κ23p22 + κ24p

32)

(κ11 + κ12p2 + κ13p21 + κ14p3

1 + p41)

2(κ21 + κ22p2 + κ23p22 + κ24p3

2 + p42)

2.

In principle, D can adopt plus and minus signs and a Hopf bifurcation mayoccur in addition to the one dimensional bifurcation of act-act systems.The system sustains one or three stationary states. The state at the originis always stable. Then it passes through two bifurcations as a function ofthe auxiliary variable s. The first of them is a saddle-node bifurcation ats = soneD, which is found in all cooperative activation-activation systems andwhere two more states are formed. The new stable state moves outwards inthe positive quadrant, i.e. to larger values of p1 and p2, and the unstablestate moves inwards. As shown in figure 18 the system passes indeed a Hopfbifurcation at s = sHopf and a stable limit cycle is formed.

The second example of intermediate regulation combines repression(F1(p2)

)and an active intermediate complex

(F2(p1)

). Here G1 and the

‘intermediate 2’ H(2)2 are the active transcription forms. The two regulatory

functions are given by

F1(p2) =κ21

κ21 + κ22p2 + κ23p22 + κ24p3

2 + p42

and

F2(p1) =κ13 p2

1

κ11 + κ12p2 + κ13p21 + κ14p3

1 + p41

.

36

Figure 18: Position and stability of the fixed points in the two-gene

cooperative system with intermediate activation and a Hill coefficient

of n = 4. The active entities are H(4)1 and H

(2)2 . A saddle-node bifurcation

s = 2.023 and a Hopf bifurcation s = 3.671 are observed. Choice of parameters:kQ

1 = kQ

2 = 2 · s, κ11 = κ12 = . . . = κ24 = 0.5/s, and kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 =d P

2 = 1. Color code see ‘Notation’.

37

Figure 19: Position and stability of the two fixed points in the two-gene

cooperative system with intermediate repression and Hill coefficient

n = 4. The active entities are G1 and H(2)2 . A pitchfork bifurcation bifurcation at

s = 3.834 and a Hopf bifurcation at s = 17.96 are observed. Choice of parameters:kQ

1 = kQ

2 = 1·s, κ11 = κ12 = . . . = κ24 = 1, and kP

1 = kP

2 = dQ

1 = dQ

2 = d P

1 = d P

2 = 1.Color code see ‘Notation’.

Computation of the regulatory determinant D yields now

D(p1, p2) = kQ

1 kQ

2 kP1kP

2 ·

· κ13κ21p1 (2κ11 + κ12p1 − κ14p31 − 2p4

1) (κ22 + 2κ23p2 + 3κ24p22 + 4p3

2)

(κ11 + κ12p2 + κ13p21 + κ14p3

1 + p41)

2(κ21 + κ22p2 + κ23p22 + κ24p3

2 + p42)

2.

Again, plus and minus signs are possible and as documented in figure 19two bifurcations, the pitchfork bifurcation which is typical for cooperativerep-rep systems and a Hopf bifurcation are indeed observed.

Without showing details we mention that the Hopf bifurcation was notobserved for all systems with regulatory determinants D that can have plusor minus sign. For example, no oscillations are observed in the systemstranscribing H

(4)1 and ‘intermediate 1’ H

(1)2 or G1 and ‘intermediate 1’ H

(1)2 ,

respectively. More examples are listed in table 5. The situation in these casesis analogous to the activation-repression system with Hill coefficient n = 2:The regulatory determinant D adopts positive values but does not exceedD = DHopf for all tested values and approached limits.

38

5 Numerical sampling of parameter space

The results derived from selected examples are summarized and augmentedby numerical explorations of parameter space in this section. Numerical sam-pling was performed in explorative manner in order to learn whether or notmore complicated cases exist where the computer assisted analytic approachapplied here is doomed to fail. For the sampling approach we assumed aconstant total gene concentration of g0 = 1 in problem adapted arbitraryunits (See section 6). All rate and equilibrium parameters, πk, k = 1, 2, . . . ,were allowed to adopt values in the range −9.25 ≤ log πk ≤ 9.25 correspond-ing to approximately 10−4 ≤ πk ≤ 104. Individual values were sampled bymeans of a random number generator assuming uniform distribution on thelogarithmic scale. A typical sample consisted of some ten thousand pointsand the distribution of different dynamical behaviors was evaluated by simplefrequency counting. The results are summarized in tables 4 and 5. Despiterelatively small samples all qualitative forms of dynamic behavior were de-tected by the numerical sampling procedure.

Table 4 presents also an overview over all activation and repression caseswith simple Hill-type functions (5) for n = 1, 2, 3, and 4. Basal activation orleaky transcription were also included. Almost all systems under considera-tion show an unregulated and a regulated regime separated by a bifurcation.The only exceptions are non-cooperative systems and the act-rep systemwith Hill coefficient n = 2.

Not all intermediate cases were investigated but the examples shown intable 5 are representative. They show transitions from one combination ofbasic regulatory scenario to another, for example from act-rep to act-act inthe table. The pure combinations have their defined scenario, Hopf bifur-cation and undamped oscillations for act-rep and one dimensional bifurca-tion and bistability for act-act and rep-rep. The intermediate cases form asmooth transition in the sense that they combine both scenarios, first onedimensional bifurcation and second Hopf bifurcation (One example combin-ing both scenarios is shown in figure 18). The best studied and documentedexample is the case for Hill coefficient n = 4 in the table. All five objectsfrom the naked gene to the complex with four monomers bound to DNAthat can possibly initiate transcription are considered in the table. The puresystems show oscillations or bistability (with one state being the origin) andthe intermediate complexes combine both behaviors – ‘intermediate 1’ and‘intermediate 2’ – or they behave like the act-act system – ‘intermediate 3’.

A very similar situation is encountered for the repression-intermediatesystem. We cannot give the details here, but figure 19 shows a parametric plotfor one intermediate case, G1 and H

(2)2 that exhibits both, a one dimensional

bifurcation leading to bistability with one state having one gene active andthe other one silenced and the second state vice versa. Later, with increasingϑ/K ratios one state becomes unstable and gives birth to a stable limit cyclevia a Hopf bifurcation.

39

Table 4: Results of bifurcation analysis of systems with simple binding functions (5) for activation and repression. Equilibriumpoints were computed by means of equations (14) and the regulatory determinant D (21) was used for stability analysis.

Regulation G1 Regulation G2 Dynamical Number of non- Bifurcation

Type F1(p2) n1 Type F2(p1) n2 pattern a negative roots b type

activation p2

K2+p21 activation p1

K1+p11 E |S 1 | 2 (2) transcritical

activation p2

K2+p21 repression K2

K1+p11 S 1 (2)

repression K2

K2+p21 repression K2

K1+p11 S 1 (2)

activationp2

2

K2+p2

2

2 activationp2

1

K1+p2

1

2 E |B(E,S) 1 | 3 (5) saddle-node

activationp2

2

K2+p2

2

2 repression K1

K1+p2

1

2 S 1 (5)

repression K2

K2+p2

2

2 repression K1

K1+p2

1

2 S |B(S1,S2) 1 | 3 ( 5) pitchfork or saddle-node

activationp3

2

K2+p3

2

3 activationp3

1

K1+p3

1

3 E |B(E,S) 1 | 3 (9) saddle-node

activationp3

2

K2+p3

2

3 repression K1

K1+p3

1

3 S |O 1 | 1 (9) Hopf

repression K2

K2+p3

2

3 repression K1

K1+p3

1

3 S |B(S1,S2) 1 | 3 (9) pitchfork or saddle-node

activationp4

2

K2+p4

2

4 activationp4

1

K1+p4

1

4 E |B(E,S) 1 | 3 (17) saddle-node

activationp4

2

K2+p4

2

4 repression K1

K1+p4

1

4 S |O 1 | 1 (17) Hopf

repression K2

K2+p4

2

4 repression K1

K1+p4

1

4 S |B(S1,S2) 1 | 3 (17) pitchfork or saddle-node

basal+activation γ1 + p2

K2+p21 basal+activation γ2 + p1

K1+p11 S 1 (2)

basal+activation γ1 +p2

2

K2+p2

2

2 basal+activation γ2 +p2

1

K1+p2

1

2 S |B(S1,S2) |S 1 | 3 | 1 (5) saddle-node | saddle-node

a The sequence of states is obtained by increasing (kQ

1 , kQ

2 ) at constant values of the other parameters, states are separated by |, and the dynamical patterns arecharacterized by the following symbols: E≡ stable fixed point at the origin P (0, 0) corresponding to both genes silenced, S≡ stable fixed point in the positivequadrant, p1 > 0, p2 > 0, B(Pi, Pj)≡ two stable fixed points separated by a saddle, and O≡ limit cycle.

b Numbers of observed fixed points with p1 ≥ 0, p2 ≥ 0 before and after the bifurcations are separated by |, the number in parentheses is the degree of the polynomialderived from equations (14). It is tantamount to the maximal number of fixed points.

40

Table 5: Results of numerical sampling of systems with randomly chosen parameter values.a

Regulation G1 Regulation G2 Dynamical Bifurcation Frequencies

Type F1(p2) n1 Type F2(p1) n2 pattern b type of patterns b

activationp2

2

κ21+κ22p2+p2

2

2 repression κ11

κ11+κ12p1+p2

1

0/2 S 1

activationp2

2

κ21+κ22p2+p2

2

2 intermediate κ12p1

κ11+κ12p1+p2

1

1/2 E |B(E,S) saddle-node 0.603|0.397

activationp2

2

κ21+κ22p2+p2

2

2 activationp2

1

κ11+κ12p1+p2

1

2/2 E |B(E,S) saddle-node 0.663|0.337

activation p23

κ21+κ22p2+κ23p2

2+p23 3 repression κ11

κ11+κ12p1+κ13p2

1+p13 0/3 S |O Hopf 0.998|0.002

activation p23

κ21+κ22p2+κ23p2

2+p23 3 intermediate κ12p1

κ11+κ12p1+κ13p2

1+p13 1/3 E |B(E,S) |O saddle-node |Hopf 0.642|0.354|0.004

activation p23

κ21+κ22p2+κ23p2

2+p23 3 intermediate

κ13p2

1

κ11+κ12p1+κ13p2

1+p13 2/3 E |B(E,S) |O saddle-node |Hopf 0.715|0.283|0.002

activation p23

κ21+κ22p2+κ23p2

2+p23 3 activation p13

κ11+κ12p1+κ13p2

1+p13 3/3 E |B(E,S) saddle-node 0.720|0.280

activationp4

2

κ21+κ22p2+κ23p2

2+κ24p3

2+p4

2

4 repression κ11

κ11+κ12p1+κ13p2

1+κ14p3

1+p4

1

0/4 S |O Hopf 0.988|0.012

activationp4

2

κ21+κ22p2+κ23p2

2+κ24p3

2+p4

2

4 intermediate κ12p1

κ11+κ12p1+κ13p2

1+κ14p3

1+p4

1

1/4 E |B(E,S) |O saddle-node |Hopf 0.673|0.321|0.006

activationp4

2

κ21+κ22p2+κ23p2

2+κ24p3

2+p4

2

4 intermediateκ13p2

1

κ11+κ12p1+κ13p2

1+κ14p3

1+p4

1

2/4 E |B(E,S) |O saddle-node |Hopf 0.740|0.258|0.002

activationp4

2

κ21+κ22p2+κ23p2

2+κ24p3

2+p4

2

4 intermediateκ14p3

1

κ11+κ12p1+κ13p2

1+κ14p3

1+p4

1

3/4 E |B(E,S) saddle-node 0.751|0.249

activationp4

2

κ21+κ22p2+κ23p2

2+κ24p3

2+p4

2

4 activationp4

1

κ11+κ12p1+κ13p2

1+κ14p3

1+p4

1

4/4 E |B(E,S) saddle-node 0.742|0.258

a The frequencies of bifurcation patterns are derived from a large numbers (N > 10 000) of randomly chosen combinations of parameters. The values for a parameterπ are taken from the interval −9.25 ≤ log π ≤ 9.25 under the assumption of a uniform distribution of log π (See also section 5).

b The sequence of states is obtained by increasing (kQ

1 , kQ

2 ) at constant values of the other parameters, states are separated by |, and the dynamical patterns arecharacterized by the symbols described in the footnote of table 4.

41

6 Discussion

The work reported here aims at the presentation of a straightforward andfairly simple mathematical technique that allows for full characterization ofthe dynamical patterns for gene regulation by transcription kinetics follow-ing equations (11) and (12). The procedure can be extended to investiga-tions of the entire parameter space since, despite a rather large number ofkinetic parameters and equilibrium binding constants, only very few quan-tities determine the dynamical pattern of the system – as encapsulated inthe fixed points and their stabilities: Stationary mRNA concentrations (15)are proportional to stationary protein concentrations (14) and therefore it issufficient to study the dynamical systems in the protein subspace. Moreover,the fixed points in protein space depend only on the binding function andone rational expression of kinetic parameters, ϑj = (kQ

j · kP

j )/(dQ

j · d P

j ), forevery gene. The stationary protein concentrations are obtained as solutionsof polynomials. Since the polynomials are of high degrees for cooperativesystems (Hill coefficient n ≥ 2) a combined analytical and computationaltechnique, consisting in the numerical calculation of the polynomial roots,is mandatory. Despite high degrees the polynomials allow for an analyticalhandling of limits like high and low binding affinities. Local stability analy-sis is performed in terms of the eigenvalues of the Jacobian matrix at fixedpoints.

In this contribution we presented examples for a (quite general) classof genetic regulations – denoted here as simple – where the analysis of theJacobian is largely facilitated by its structure (21): Computation of the reg-ulatory determinant D is sufficient for the stability analysis of fixed points.At two computable critical values, DoneD and DHopf , the fixed points be-comes unstable through a one dimensional bifurcation or a Hopf bifurcation,respectively. Fixed points are stable in between, DoneD < D < DHopf . Itis important for generalizations that DoneD is always negative whereas DHopf

has always positive sign. For two gene systems the class simple is constitutedby cross-catalysis which expresses that regulatory binding functions dependonly on the concentration of the protein derived from the other gene: F1(p2)and F1(p2). Apart from this restriction the binding functions can be arbitrar-ily complicated, only differentiability is required for the computation of D:Examples for more complicated cases analyzed here are leaky transcription(two terms) and intermediate complexes as transcription initiators.

The approach can be readily extended to more than two genes and therethe properties of the class simple are fulfilled by catalytic cycles consistingof a closed loop of regulatory functions, GN ⇒ G1 ⇒ G2 ⇒ · · · ⇒ GN forarbitrary N . A description of such regulatory systems by means of dynamicalgraphs has been reported in [32]. One concrete example of a regulatory loopwith N = 3 is the repressilator [15] which has been analyzed in detail (forexample in [33]). On the other hand, there are also examples of two genesystems that do not fall under the classification simple, for example self-activation and cross-repression or self-repression and cross activation, becausethen the regulatory binding functions depend on the concentrations of bothproteins: F1(p1, p2) and F2(p1, p2). Attempts to generalize our approach andto group these non-simple systems into subgroups according to the dynamical

42

structure related to the difficulty of analysis are under way [27].The dynamical pattern of gene regulation has been analyzed for sev-

eral cooperative binding functions, F1(p2) and F2(p1), with different Hillcoefficients by means of a new technique using the regulatory determinantD(p1, p2) introduced and defined in equation (21). We computed and clas-sified only the generic dynamic features and the bifurcation patterns whichwere found in full agreement with the literature wherever previous studieswere. No attempt has been made yet to make a complete search in parameterspace, nor did we try in this paper to adjust to experimental data. Thereforeconcentrations and parameter values were chosen to illustrate best the basicfeatures of the bifurcation diagrams and the oscillatory dynamics. A forth-coming study will deal with fitting regulatory dynamics to experimental databy making use of inverse methods [34, 35]. A particularly challenging prob-lem is reverse engineering of bifurcation patterns for which first approachesare now available [36].

All cooperative systems (except activation-repression with Hill coefficientn = 2) show an unregulated and a regulated regime. The regulated regimeis reached at sufficiently high values of the ratio ϑ/K implying that (i) tran-scription and translation are fast enough compared to mRNA and proteindegradation, and (ii) binding is sufficiently strong. The pure systems10 fallinto three classes: act-act, act-rep, and rep-rep. Each class has its ownregulatory characteristic, (i) act-act leads to both genes active or both genessilenced, (ii) act-rep results in oscillatory activity of the two genes, and (iii)rep-rep represents a bistable switch with the two states: (i) G1 active andG2 silenced, and vice versa (ii) G1 silenced and G2 active. In pure systems Dis either always negative (act-act and rep-rep) or always positive (act-rep)and accordingly we find bistability only in the first two classes of systemsand oscillations occur exclusively in the third class. More complicated bind-ing functions may give rise to mixed behavior resulting from simultaneousappearance of one dimensional and Hopf bifurcations in the same bifurcationdiagram. The cases of intermediate regulation discussed in section 4.2 mayserve as examples.

The model for gene regulation and the technique to analyze regulatorydynamics presented here provides a fast tool for computational bifurcationanalysis. The parameter spaces of small genetic networks with several genescan be scanned completely. Regulatory systems can be classified into simple

and non-simple systems according to the structure of the Jacobian matrix.Simple systems are accessible to a highly efficient combined analytical andcomputational approach. Analytical expressions are available for the compu-tation of bifurcation points. The current procedure will be developed furtherinto an automatic tool for the exploration of entire parameter spaces that isapplicable to systems with several genes (approximately up to five genes).Future work aims also at an upscaling to systems with many genes.

10The term pure indicates that the complex active in transcription is either fullysaturated – H(4) in figure 17 implying activation (act) – or unbound – G in figure 17indicating repression (rep).

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Acknowledgements

This work has been supported financially by the Austrian ’Fonds zurForderung der wissenschaftlichen Forschung’ (FWF, Projects: P-13887 andP-14898). It is also part of the project on ’Inverse Methods in Biology andChemistry’ sponsored by the ’Wiener Wissenschafts-, Forschungs- und Tech-nologiefonds’ (WWTF, Project: MA05). Part of the work has been carriedout during a visit at the Santa Fe Institute within the External FacultyProgram. All support as well as fruitful discussions with Professors KarlSigmund, Josef Hofbauer and Heinz Engl, Drs. Christoph Flamm, James Lu,and Stefan Muller, and Mag. Lukas Endler are gratefully acknowledged.

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Appendix: Condition for a Hopf bifurcation

In order to compute the value DHopf (figure 2) where the two-gene systemlooses stability at positive values of D, we use the criterion by Lienard-Chipart (see [37], pp.221). According to that criterion, the eigenvalues of theJacobian have strictly negative part if and only if the zeroth, second, andfourth coefficient of the secular equation as well as the second and fourthHurwitz determinant are positive. The latter two are determinants of 2 × 2and 4 × 4 matrices, respectively, whose nonzero entries are coefficient of thesecular equation.

From equation (21), the zeroth and the second coefficient are always pos-itive, and the fourth coefficient is positive for D > −dQ

1 dQ

2 d P

1 d P

2 . The secondHurwitz determinant is always positive because it expands to an expressionin dQ

1 , dQ

2 , d P

1 , d P

2 with positive coefficients. The fourth Hurwitz determinant isa quadratic polynomial in D. With the help of the computer algebra systemMaple, one finds that it has two real roots, corresponding to the values

Dtrans = − dQ

1 dQ

2 d P

1 d P

2 and (23)

DHopf =(dQ

1 + dQ

2 )(dQ

1 + d P

1 )(dQ

1 + d P

2 )(dQ

2 + d P

1 )(dQ

2 + d P

2 )(d P

1 + d P

2 )

(dQ

1 + dQ

2 + d P

1 + d P

2 )2. (25)

Between these two roots it is positive because its leading coefficient is neg-ative, the eigenvalues of the Jacobian have strictly negative real parts, andthe corresponding fixed point is asymptotically stable.

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